On 2010 AKHMEDOV and PARK claimed there are infinitely many exotic smooth structures on $S^2\times S^2$, see http://arxiv.org/abs/1005.3346
Then Rasmussen posted a paper : Perfect Morse functions and exotic $S^2 \times S^2$'s http://arxiv.org/abs/1005.4674 But he withdrawn his paper, here is his note:
The main theorem of the paper shows that a smooth manifold which is homeomorphic to S^2xS^2 and has nonvanishing Ozsvath-Szabo invariant does not admit a perfect Morse function. I am withdrawing the paper because it is unclear to me if such a manifold exists.
Did I understand correctly that the first paper mentioned above had a gap? Which means one don't know whether there exists exotic smooth structure on $S^2\times S^2$.